Bilinear Forms Graphs

• Bilinear forms graphs are algebraic-combinatorial structures defined via low-rank differences of matrices over finite fields that reveal explicit spectral and regularity properties.
• They exhibit distance-regularity with well-defined parameters such as intersection arrays and eigenvalues, enabling a thorough classification of maximal cliques and symmetry analysis.
• Their versatile models and geometric realizations support applications in coding theory, design geometry, operator theory, and representation theory.

A bilinear forms graph is a highly structured algebraic-combinatorial object defined via the geometry of low-rank differences between matrices over a finite field or ring. Its study connects combinatorics, finite geometry, algebraic graph theory, and applications such as coding theory. The canonical form is the graph whose vertices are $m \times n$ matrices over a finite field $\mathbb{F}_q$, with adjacency given by single-rank differences. This structure exhibits rich regularity, explicit spectral properties, and a classification of maximal cliques closely linked to classical objects in design and geometry.

1. Formal Definition and Basic Construction

Let $\mathbb{F}_q$ denote the finite field of order $q$, and $M_{m \times n}(\mathbb{F}_q)$ the set of $m \times n$ matrices over $\mathbb{F}_q$. The bilinear forms graph, denoted $\mathrm{Bil}_q(m,n)$, is the simple, undirected graph with:

• Vertices: All $m \times n$ matrices over $\mathbb{F}_q$.
• Edges: Two distinct matrices $A, B$ are adjacent if and only if $\mathrm{rank}(A-B) = 1$.

This construction generalizes to more exotic rings such as residue class rings $\mathbb{Z}_{p^s}$, where the adjacency is defined using the Cohn-inner-rank (least $r$ such that $A-B = B'C'$ for $B' \in R^{m \times r},~C' \in R^{r \times n}$) (Huang, 2017, 2002.03560). The field case ($R = \mathbb{F}_q$) forms a distance-regular, highly symmetric object.

2. Combinatorial Structure and Parameters

General Parameters

The principal combinatorial characteristics of $\mathrm{Bil}_q(m,n)$ are:

• Order: $v = q^{mn}$
• Degree (Valency): For $A\in V$, the number of rank-1 matrices (excluding $0$) is $k = \frac{(q^m-1)(q^n-1)}{q-1}$ (Flórez et al., 15 Jul 2024, Terwilliger et al., 13 Dec 2025).
• Diameter: For $m \leq n$, the diameter is $m$.

For special cases, notably $m=2$ or $n=2$, the graph is strongly regular, with

• $\lambda = q^n + q^2 - q - 2$
• $\mu = q(q+1)$

as parameters (Flórez et al., 15 Jul 2024).

Distance-Regularity and Intersection Arrays

When $m = n = d$ and $q=2$, the graph $\mathrm{Bil}_2(d\times d)$ is distance-regular of diameter $d$ (Gavrilyuk et al., 2015). The intersection array is given explicitly as:

$$\left\{ 2^d-1,\; 2^2(2^{d-1}-1),\; 2^4(2^{d-2}-1),\; \dots,\; 2^{2d-2}(2^1-1)\;; 1,2,2^2,\dots,2^{d-1} \right\}$$

The intersection numbers are: $$b_{i-1} = 2^{2i-2}(2^{d-i+1}-1),\quad c_i=2^{i-1},\quad 1 \leq i \leq d.$$ This array fully characterizes the graph in this regime: any distance-regular graph with the same intersection array is isomorphic to $\mathrm{Bil}_2(d\times d)$ for $d \geq 3$ (Gavrilyuk et al., 2015).

3. Eigenstructure and Symmetry

The eigenvalues for the adjacency matrix are determined by the classical parameters. For $\mathrm{Bil}_2(d\times d)$, the eigenvalues are

$$\theta_0 = 2^d - 1,\qquad \theta_i = 2^{d-i} - 1 - 2^i\quad (i=1,\dots,d)$$

with explicit multiplicities: $m_i = \binom{d}{i}_2 \prod_{j=1}^i (2^j - 1)$ where $\binom{d}{i}_2$ is the $q$-ary Gaussian binomial coefficient (Gavrilyuk et al., 2015).

The graph is vertex-transitive (as a Cayley graph for the additive group of $M_{m \times n}(\mathbb{F}_q)$), and, in the parameters above, distance-regular or strongly regular. The automorphism group is large, containing $$\mathrm{GL}_m(\mathbb{F}_q) \times \mathrm{GL}_n(\mathbb{F}_q)$$ (Flórez et al., 15 Jul 2024).

4. Maximal Cliques and Geometric Realizations

Maximal cliques fall into two canonical types:

• Row-type cliques: Consist of all matrices sharing all but a fixed row, forming sets of size $q^n$ (for $m \leq n$).
• Column-type cliques: Consist of all matrices sharing all but a fixed column, forming sets of size $q^m$ (for $n \leq m$).

In the case $m = n$, both forms exist. In generalized versions over rings, more complex clique types appear, governed by Smith normal form invariants (Huang, 2017, 2002.03560).

In the projective-rectangle interpretation, maximal cliques correspond to families of lines through a point (point-cliques) or lines contained in a subplane (plane-cliques) in the associated incidence geometry (Flórez et al., 15 Jul 2024).

5. Algebraic and Geometric Models

The bilinear forms graph has multiple equivalent realizations:

• Matrix Model: Vertices are $m \times n$ matrices over $\mathbb{F}_q$, adjacency by rank-1 difference (Gavrilyuk et al., 2015, Flórez et al., 15 Jul 2024).
• Subspace Model: Vertices are $m$-dimensional subspaces of a $V = \mathbb{F}_q^{m+n}$ disjoint from a fixed $n$-dimensional subspace; adjacency if intersection is $(m-1)$-dimensional (Feng et al., 2011).
• Projective Geometry Model: For $m=2$, lines of a projective "rectangle" whose intersections realize the graph, with parameters matching those of $\mathrm{Bil}_q(2, n)$ (Flórez et al., 15 Jul 2024).
• Quiver/Algebraic Models: Incidence matrices of marked ribbon graphs associated to gentle algebras yield Gram matrices for homological bilinear forms, embedding the combinatorics in the homological algebra and representation theory context (González et al., 5 Jul 2024).
• Dirichlet Forms: Certain separable quadratic forms can be approximated by energy forms on (weighted) graphs structurally related to bilinear forms graphs (Hinz et al., 2015).

In all cases, the underlying organizing principle is the interaction of low-rank linear structure and combinatorial adjacency.

6. Extensions, Partitions, and Algebraic Modules

Several research developments exploit or generalize the structure:

• Generalized Bilinear Forms Graphs: Extending to graphs over residue class rings ($\mathbb{Z}_{p^s}$ or general $\mathbb{Z}_h$), with adjacency by inner rank less than $d$, enabling applications to MRD and MDS codes, and yielding new extremal set characterizations—e.g., generalized Erdős–Ko–Rado theorems for $r$-intersecting matrix families (Huang, 2017, 2002.03560).
• Equitable Partitions and Subconstituent Algebras: For distance-regular bilinear forms graphs, equitable partitions (such as the $(x, y)$-partition indexed by distance to a fixed edge) can be constructed, leading to modules for the Terwilliger algebra that decompose into irreducible summands described by explicit spectral data. For diameter $D$, the equitable $(x,y)$-partition yields $6D-2$ cells and a $T$-module $U$ decomposing into five irreducible modules (one primary, four of endpoint one) (Terwilliger et al., 13 Dec 2025).
• Metric Dimension: The metric dimension—minimum cardinality of a resolving set—has sharp upper bounds, with explicit constructions using partitioning of the ambient vector space. For $B_q(n,m)$, $\beta(B_q(n,m)) \leq q^{n+m-1}$ when $n \geq m+2$, and $\leq q^{n+m}$ otherwise, outperforming more general combinatorial bounds (Feng et al., 2011).

7. Applications and Interpretations

• Coding Theory: The structure of independent sets and cliques in generalized bilinear forms graphs encodes maximal rank distance (MRD) codes and MDS codes over finite rings and fields. Every maximum independent set is an MRD code, meeting the Singleton bound (Huang, 2017).
• Convex Hulls and Polyhedral Theory: For the graph of a bilinear function's values over $[0,1]^n$, extended formulations leveraging the combinatorics of cycles, cliques, and almost-cliques yield concise characterizations of the convex hull, linking polyhedral combinatorics and algebraic graph theory (Gupte et al., 2017).
• Partial Geometries: In the projective rectangle interpretation, the bilinear forms graph realizes an incidence geometry with associated partial geometry parameters, with maximal cliques furnishing the lines of such structures (Flórez et al., 15 Jul 2024).
• Dirichlet Forms and Operator Theory: Separable quadratic forms on function algebras admit approximations by energy forms on finite graphs structurally analogous—with discrete forms converging in the strong resolvent sense to the continuous limit—highlighting the analytical import of bilinear forms graphs in potential theory (Hinz et al., 2015).
• Representation Theory: In the context of $W$-graph representations of Iwahori–Hecke algebras, explicit computation of invariant bilinear forms (Gram matrices) employs linear algebra over integral domains, relating to the graph-theoretic underpinnings in spectral theory (Geck et al., 2017).

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The bilinear forms graph thus serves as a central object in algebraic combinatorics, with deep connections to group actions, partial geometries, coding theory, operator theory, and the structure of combinatorial extremal families. Its unique characterization via intersection array for key parameter regimes, coupled with its algebraic models and geometric representations, provides a unified framework in discrete mathematics and theoretical computer science (Gavrilyuk et al., 2015, Huang, 2017, Flórez et al., 15 Jul 2024, Terwilliger et al., 13 Dec 2025, Feng et al., 2011, Hinz et al., 2015, Gupte et al., 2017, González et al., 5 Jul 2024).